On the Size of K-Cross-Free Families

Kupavskii, A. and Pach, János and Tomon, István (2018) On the Size of K-Cross-Free Families. COMBINATORICA. ISSN 0209-9683

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Abstract

Two subsets A,B of an n-element ground set X are said to be crossing, if none of the four sets A∩B, A\B, B\A and X\(A∪B) are empty. It was conjectured by Karzanov and Lomonosov forty years ago that if a family F of subsets of X does not contain k pairwise crossing elements, then |F|=O(kn). For k=2 and 3, the conjecture is true, but for larger values of k the best known upper bound, due to Lomonosov, is |F|=O(knlogn). In this paper, we improve this bound for large n by showing that |F|=Ok(nlog*n) holds, where log* denotes the iterated logarithm function. © 2018 János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature

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